等比級數

小PK

1.一無窮等比級數的前10項和為40，前20項和為64，求此無窮等比級數的和?

2.試求
0.2+0.022+0.00222+.............之和?


[ 本文最後由 M.N.M. 於 06-12-3 08:41 AM 編輯 ]

M.N.M.

1.
a[1-(r^10)] / (1-r) =40......(1)

a[1-(r^20)] / (1-r) =64......(2)

(1)/(2)

=>[1-(r^10)] / [1-(r^20)] = 40/64=5/8

8-8(r^10)=5-5(r^20)

5(r^10)^2 -8(r^10)+3=0

r^10=1 or 3/5 (1不合，題意不合)


設a/(1-r)=k......(3)

(1)/(3)

=>1-(r^10)=40/k

2/5=40/k

k=100



2.
0.9+0.099+0.00999+...

=(1-0.1)+(0.1-0.001)+(0.01-0.00001)+...

=(1+0.1+0.01+...)-(0.1+0.001+0.00001+...)

=(1/0.9)-(0.1/0.99)

=100/99

0.2+0.022+0.00222+...

=(2/9)(0.9+0.099+0.00999+...)

=(2/9)(100/99)

=200/891